Draft analysis¶

Group name: Gruppe B Raghid Allaham & Niels-Janos Simmet

Introduction¶

Over the last couple of years American Football gained a lot of popularity in Europe especially Germany. Different than soccer, American football is a play based game. After each play the game gets interupted for the offense and defense to set up their team once again. The goal is to achieve a touchdown on the opponents endzone. To achieve that the offense works its way across the field play by play. With each play the offense tries to gain as many yards as possible while the defense tries to prevent this. There are 2 main kind of plays to gain yards: rushing and passing. For a rush plays the offensive team tries to gain yards by caring the ball. For pass plays the quarterback of the offensive team will throw to ball to another player. The key difference is the that for rushing plays the ball will not be airborn but handed over.

We want to find out if there is a way to predict how many yards a team will gain based on variables that we know before the play starts.

Other people have already researched similiar questions but i.e. focused on predicting the rushing yards. We focus on yards generally gained no matter if rushed or passed, as this can not be determined before the play starts. Interesting links to check out for similiar analysis:

http://cs229.stanford.edu/proj2019aut/data/assignment_308832_raw/26588266.pdf

https://rpubs.com/woutcault/Final606_Multiple_Regression

https://medium.com/@matthewdmeans/predicting-yards-gained-in-the-nfl-3a0eea7a54a3

Data Description¶

For NFL matches there is a comprehensive data asset available that includes play-by-play data, we will focus on the data gathered so far (until 26th December 2022) in the 2022 NFL season. It tracks a multitude of variables for each play. For our analysis we will focus on the variables that are know before the play starts as explanatory variables and the yards gained as the response variable. The data was scraped from the NFL Next Gen Stats website with the NFL verse package.

Data Source https://github.com/nflverse/nflverse-data/releases/tag/pbp

Data Description https://mrcaseb.github.io/pages_dummy/reference/fast_scraper.html

Our response variable will be "yards-gained" this variable describes how many yards a team gained (or lost) in a play excluding yards gained via fumble recoveries and laterals.

Our possible explanatory variables are:¶

"quarter_seconds_remaining" - Seconds on the clock until the quarter ends

"qtr" - Current quater 1-4 and 5 for overtime

"down" - Current down

"yardline_100" - Distance to the opponents endzone in yards

"ydstogo" - Yards to the next first down

"score_differential" - Score difference between offense team and the defense team

"posteam" - Offense Team

"defteam" - Defense Team

"home_team" - Home Team

"away_team" - Guest/Away Team

"weather" - Weather conditions

Data Dictionary¶

In [ ]:
#Import libraries
import pandas as pd
import data_dic 

#Create Data Frame
data_dictionary_content = data_dic.data_dictionary_content
data_dictionary = pd.DataFrame(data_dictionary_content)
data_dictionary
Out[ ]:
Name Description Role Type Format
0 yards_gained Yards gained/lost in the play excluding yards ... Response numeric float64
1 quarter_seconds_remaining Seconds on the clock until the quarter ends Predictor numeric int64
2 qtr Current quater 1-4 and 5 for overtime Predictor numeric int64
3 down Current down Predictor numeric float64
4 yardline_100 Distance to the opponents endzone in yards Predictor numeric float64
5 ydstogo Yards to the next first down Predictor numeric int64
6 score_differential Score difference between offense team and the ... Predictor numeric float64
7 posteam Offense Team Predictor nominal object
8 defteam Defense Team Predictor nominal object
9 home_team Home Team Predictor nominal object
10 away_team Guest/Away Team Predictor nominal object

Data Setup¶

In [ ]:
#Import libraries
import altair as alt
import numpy as np
import matplotlib.pyplot as plt
import time
import datetime
import pickle
from sklearn.linear_model import LinearRegression
from sklearn.neighbors import KNeighborsRegressor
from sklearn.preprocessing import StandardScaler
from sklearn.linear_model import Lasso
from sklearn.metrics import mean_squared_error
from sklearn.metrics import mean_absolute_error
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LassoCV
import warnings

alt.data_transformers.disable_max_rows()
warnings.simplefilter(action='ignore', category=FutureWarning)

# Set Pandas to show more rows in the output
pd.set_option('display.max_rows', 500)

Import data¶

In [ ]:
# Loading Data
path = "..\\data\\raw\\play_by_play_2022.csv"
df = pd.read_csv(path, low_memory=False)

Data corrections¶

We want to find out if there is a way to predict how many yards a team will gain based on variables that we know before the play starts. So this small list represents the variables, which are already known before a play:

In [ ]:
# Select the relevant variables
column_selection = [
"yards_gained",
"home_team",
"away_team",
"quarter_seconds_remaining",
"qtr",
"down",
"yardline_100",
"ydstogo",
"posteam",
"defteam",
"score_differential"
]
# Filter dataframe for relevant variables
df_small = df[column_selection]
# drop rows with missing values
df_small.dropna(inplace=True)

C:\Users\rahgi\AppData\Local\Temp\ipykernel_15756\1268166200.py:20: SettingWithCopyWarning: 
A value is trying to be set on a copy of a slice from a DataFrame

See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
  df_small.dropna(inplace=True)

Data structure¶

In [ ]:
df_small.info()

<class 'pandas.core.frame.DataFrame'>
Int64Index: 35058 entries, 2 to 42322
Data columns (total 11 columns):
 #   Column                     Non-Null Count  Dtype  
---  ------                     --------------  -----  
 0   yards_gained               35058 non-null  float64
 1   home_team                  35058 non-null  object 
 2   away_team                  35058 non-null  object 
 3   quarter_seconds_remaining  35058 non-null  int64  
 4   qtr                        35058 non-null  int64  
 5   down                       35058 non-null  float64
 6   yardline_100               35058 non-null  float64
 7   ydstogo                    35058 non-null  int64  
 8   posteam                    35058 non-null  object 
 9   defteam                    35058 non-null  object 
 10  score_differential         35058 non-null  float64
dtypes: float64(4), int64(3), object(4)
memory usage: 3.2+ MB

Variable lists¶

In [ ]:
# define outcome variable as y_label
y_label = 'yards_gained'
# select features and drop y-variable
X = df_small.drop(['yards_gained'], axis=1)
# create response
y = df_small[y_label]

Data preparation¶

In [ ]:
# List of categorial predictive variables
list_categorial = [
"posteam",
"defteam",
"home_team",
"away_team"
]
# List of our numeric predictive variables
list_numeric = [
"quarter_seconds_remaining",
"qtr",
"down",
"yardline_100",
"ydstogo",
"score_differential"
]
In [ ]:
# Data correction for the categorial variables
X[list_categorial] = X[list_categorial].astype("category",copy=False)
# Create dummies for the categorial variables, so we can fit them in our model
dummies = pd.get_dummies(X[list_categorial])
# convert the headers to lower_case
dummies.columns = [x.lower() for x in dummies.columns]
# concatenate the categorial variables to the numerics and get all predictive variables in X.
X = pd.concat([X[list_numeric], dummies], axis=1)

Data splitting¶

We do a train/test split with 30/70 % of the data.

In [ ]:
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, shuffle=True, random_state = 44)

Analysis¶

Descriptive statistics¶

In [ ]:
# Summary statistics
print(df_small.mean())
df_small.describe().T

yards_gained                   4.668435
quarter_seconds_remaining    424.119516
qtr                            2.574391
down                           1.997005
yardline_100                  51.043585
ydstogo                        8.464231
score_differential            -1.414114
dtype: float64
Out[ ]:
count mean std min 25% 50% 75% max
yards_gained 35058.0 4.668435 8.131966 -26.0 0.0 2.0 7.0 98.0
quarter_seconds_remaining 35058.0 424.119516 272.601532 0.0 174.0 413.0 659.0 900.0
qtr 35058.0 2.574391 1.135944 1.0 2.0 3.0 4.0 5.0
down 35058.0 1.997005 1.003640 1.0 1.0 2.0 3.0 4.0
yardline_100 35058.0 51.043585 24.274817 1.0 32.0 55.0 71.0 99.0
ydstogo 35058.0 8.464231 4.081100 1.0 6.0 10.0 10.0 38.0
score_differential 35058.0 -1.414114 9.296843 -37.0 -7.0 0.0 3.0 37.0

we have a huge amount of observation here: 35058

  • We notice that the mean of yards gained after each play is about 4.6 with std of 8.1.
  • The rest of the data only reflects the rules and characteristics of the game. #TODO

Exploratory data analysis¶

In [ ]:
# Histogram Occurency of Yards gained
df_tmp = pd.DataFrame(df_small).copy()
alt.Chart(df_tmp).mark_bar().encode(
    x=alt.X("yards_gained"),
    y=alt.Y('count()'),
).interactive() | alt.Chart(df_tmp).mark_bar().encode(
    x=alt.X("yards_gained"),
    y=alt.Y('count()',       
            scale=alt.Scale(type='symlog'))
)
Out[ ]:

Left: We can see that many plays result in zero yards gained. This is due to plays being aborted i.e. by incomplete passes or fouls.

Right: in the second diagram By making the y-axis logarithmic we can also display the histogram better and its already start looking like a normal distribution.

In [ ]:
# Histogram Occurency of Yards gained without plays were no yards were gained/lost
df_tmp=df_small.loc[df.yards_gained != 0]
alt.Chart(df_tmp).mark_bar().encode(
    x=alt.X("yards_gained"),
    y=alt.Y('count()'),
).interactive()
Out[ ]:

After filtering out the plays that result zero yards gained, we get a distribution like the one we would expect: a positive skewed normal distribution, with a mean = 4.6.

  • Probability that a team gains positive yards with a play is 57.94 %
In [ ]:
# Calculation of probability that a team gains yards with a play
print("Probability that a team gains positive yards with a play: " + str(round((len(df_small.loc[df_small.yards_gained > 0])  / len(df_small)) * 100, 2) ))

Probability that a team gains positive yards with a play: 57.94

Relationships¶

Now we want to understand the relationship between our numerical variables.

In [ ]:
# Explorative analysis of correlation between the variables
alt.Chart(df_tmp).mark_circle().encode(
    x=alt.X(alt.repeat("column"), 
            type='quantitative',
            scale=alt.Scale(zero=False)
            ),
    y=alt.Y(alt.repeat("row"), 
            type='quantitative',
            scale=alt.Scale(zero=False)
             )
).properties(
    width=150,
    height=150
).repeat(
    row = list_numeric + ["yards_gained"],
    column=list_numeric + ["yards_gained"]
)
Out[ ]:

Some correlation in the data can already be seen, most of them are game roles and restriction related:

  • yards_gaind and yardline_100: the maximum yards that can be gained are to the finish line.

Correlation Matrix¶

With a correlation matrix we want to test which of our variables correlate to each other by what amount. For that we chose the Spearman Method over the Pearson and Kendall Method. It does not really on normality of the data as the Pearson method does. And tests for correlation not dependence as the Kendall Method does

In [ ]:
# Correlation Matrix: Spearman Method
corr = df_tmp[list_numeric + ["yards_gained"]].corr(method="spearman").round(2)
corr = corr.style.background_gradient(cmap='twilight_shifted',axis=None)
corr
Out[ ]:
  quarter_seconds_remaining qtr down yardline_100 ydstogo score_differential yards_gained
quarter_seconds_remaining 1.000000 -0.040000 -0.030000 0.080000 -0.000000 0.010000 0.010000
qtr -0.040000 1.000000 0.020000 -0.040000 0.000000 -0.030000 -0.020000
down -0.030000 0.020000 1.000000 -0.080000 -0.460000 -0.010000 0.050000
yardline_100 0.080000 -0.040000 -0.080000 1.000000 0.220000 -0.010000 0.090000
ydstogo -0.000000 0.000000 -0.460000 0.220000 1.000000 -0.000000 0.090000
score_differential 0.010000 -0.030000 -0.010000 -0.010000 -0.000000 1.000000 -0.070000
yards_gained 0.010000 -0.020000 0.050000 0.090000 0.090000 -0.070000 1.000000

We can see a that most of the variables do not strongly correlate with each other.

ydstogo and down are quite strongly correlated as expected, because with each down the offensive team should come closer to the yardline it needs to reach.

ydstogo and yardline_100 also have some correlation which can be explained by the underlying rules of the game. The offensive team can not have more yards to go than they have until they enter the defenses endzone, the distance to which is represented by the yardline_100 variable. Therefor it's always ydstogo <= yardline_100

yards_gained and down also seem to have a correlation which can be explained by

  • the offensive team rather punting (kicking the ball to the direction of the defenses endzone which usually allows the defense to become offense) than trying to gain yards on 4th down
  • game tactics as the target distance ydstogo correlates with down. Teams might more often go for plays that result in just making it to the yardline on 3rd or 4th down rather than aiming for big plays that result in many yards gained.

Modeling¶

In this section we will look at 3 different regression models (Linear regression, K-nearest neighbour, & Lasso-regression) as well as Model 4 were we just always predict the mean value as a sanity check.

Model 1: Linear regression model¶

In [ ]:
# Select a linear regression model
reg = LinearRegression()
X.head()
Out[ ]:
quarter_seconds_remaining qtr down yardline_100 ydstogo score_differential posteam_ari posteam_atl posteam_bal posteam_buf ... away_team_no away_team_nyg away_team_nyj away_team_phi away_team_pit away_team_sea away_team_sf away_team_tb away_team_ten away_team_was
2 896 1 1.0 78.0 10 0.0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0
3 869 1 1.0 59.0 10 0.0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0
4 865 1 2.0 59.0 10 0.0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0
5 841 1 3.0 54.0 5 0.0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0
6 833 1 4.0 64.0 15 0.0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0

5 rows × 134 columns

In [ ]:
# Fit the model
reg.fit(X_train, y_train)
Out[ ]:
LinearRegression()
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LinearRegression()
In [ ]:
# Model intercept
reg.intercept_
Out[ ]:
4.4254004850472
In [ ]:
# Some coefficient of the Model 10/134
reg.coef_[0:10]
Out[ ]:
array([ 0.0755239 , -0.        , -0.84930098,  0.70234953,  0.03170535,
       -0.22428543, -0.06483769, -0.        ,  0.        ,  0.0805432 ])

Evaluation on Test set¶

In [ ]:
# Prediction for our data
y_pred = reg.predict(X_test)
In [ ]:
# Mean squared error model 1
model1MSE = mean_squared_error(y_test, y_pred)
model1MSE
Out[ ]:
63.52165182927777
In [ ]:
# Root mean squared error model 1
model1RMSE = mean_squared_error(y_test, y_pred, squared=False)
model1RMSE
Out[ ]:
7.9700471660635595
In [ ]:
# Mean absolute error model 1
model1MAE = mean_absolute_error(y_test, y_pred)
model1MAE
Out[ ]:
5.315078528531733

Save Model¶

Save your model in the folder models/. Use a meaningful name and a timestamp.

In [ ]:
ts = time.time()
file = '../models/linear_reg_model.sav' + datetime.datetime.fromtimestamp(ts).strftime('_%Y-%m-%d-%H_%M_%S') + ".sav"
# pickle.dump(reg, open(file, 'wb')) # uncommented to reduce dumping models, after each reload. 
file
Out[ ]:
'../models/linear_reg_model.sav_2023-01-16-00_54_03.sav'

Model 2: K-Nearest Neighbor Model¶

In [ ]:
reg2 = KNeighborsRegressor(n_neighbors=2)

reg2.fit(X_train, y_train)
Out[ ]:
KNeighborsRegressor(n_neighbors=2)
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KNeighborsRegressor(n_neighbors=2)

Evaluation on Test set¶

In [ ]:
y_pred2 = reg2.predict(X_test)
In [ ]:
# Mean squared error model 2
model2MSE = mean_squared_error(y_test, y_pred2)
model2MSE
Out[ ]:
97.9147413957026
In [ ]:
# Root mean squared error model 2
model2RMSE = mean_squared_error(y_test, y_pred2, squared=False)
model2RMSE
Out[ ]:
9.895187789814937
In [ ]:
# Mean absolute error model 2
model2MAE = mean_absolute_error(y_test, y_pred2)
model2MAE
Out[ ]:
6.549961969956265

Save Model¶

Save your model in the folder models/. Use a meaningful name and a timestamp.

In [ ]:
ts = time.time()
file = '../models/KN_Model.sav' + datetime.datetime.fromtimestamp(ts).strftime('_%Y-%m-%d-%H_%M_%S') + ".sav"
# pickle.dump(reg2, open(file, 'wb')) # uncommented to reduce dumping models, after each reload. 
file
Out[ ]:
'../models/KN_Model.sav_2023-01-16-00_54_04.sav'

Model 3: Lasso Regression¶

Standardization¶

  • Lasso performs best when all numerical features are centered around 0 and have variance in the same order.
  • If a feature has a variance that is orders of magnitude larger than others, it might dominate the objective function and make the estimator unable to learn from other features correctly as expected.
  • This means it is important to standardize our features.
$$z = \frac{(x- \bar x)}{s}$$
In [ ]:
scaler = StandardScaler().fit(X_train) 
X_train = scaler.transform(X_train)
X_test = scaler.transform(X_test)
In [ ]:
# numpy.linspace(start, stop, num of samples)
alphas = np.linspace(0.01,500,100)
lasso = Lasso(max_iter=10000)
coefs = []
for a in alphas:
    lasso.set_params(alpha=a)
    lasso.fit(X_train, y_train)
    coefs.append(lasso.coef_)
In [ ]:
ax = plt.gca()
ax.plot(alphas, coefs)
ax.set_xscale('log')
plt.axis('tight')
plt.xlabel('alpha')
plt.ylabel('Standardized Coefficients')
plt.title('Lasso coefficients as a function of alpha');

Lasso with optimal alpha¶

  • To find the optimal value of alpha, we use scikit learns lasso linear model with iterative fitting along a regularization path (LassoCV).

  • The best model is selected by cross-validation.

k-fold cross validation¶
In [ ]:
# Lasso with 5 fold cross-validation
reg = LassoCV(cv=5, random_state=0, max_iter=10000)

# Fit model
reg.fit(X_train, y_train)
Out[ ]:
LassoCV(cv=5, max_iter=10000, random_state=0)
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LassoCV(cv=5, max_iter=10000, random_state=0)
In [ ]:
# Show best value of penalization chosen by cross validation:
reg.alpha_
Out[ ]:
0.044955496590011336
In [ ]:
# Set best alpha
lasso_best = Lasso(alpha=reg.alpha_)

lasso_best.fit(X_train, y_train)
Out[ ]:
Lasso(alpha=0.044955496590011336)
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Lasso(alpha=0.044955496590011336)

Evaluation on Test set¶

In [ ]:
# Show score

print('R squared training set', round(lasso_best.score(X_train, y_train)*100, 2))
print('R squared test set', round(lasso_best.score(X_test, y_test)*100, 2))

R squared training set 2.78
R squared test set 2.29
In [ ]:
# Mean squared error model 3
model3MSE = mean_squared_error(y_test, lasso_best.predict(X_test))
model3MSE
Out[ ]:
63.42394341570026
In [ ]:
# Root mean squared error model 3
model3RMSE = mean_squared_error(y_test, lasso_best.predict(X_test), squared=False)
model3RMSE
Out[ ]:
7.963915080894588
In [ ]:
# Mean absolute error model 3
model3MAE = mean_absolute_error(y_test, lasso_best.predict(X_test))
model3MAE
Out[ ]:
5.312488557149409
In [ ]:
plt.semilogx(reg.alphas_, reg.mse_path_, ":")

plt.plot(
    reg.alphas_ ,
    reg.mse_path_.mean(axis=-1),
    "k",
    label="Average across the folds",
    linewidth=2,
)
plt.axvline(
    reg.alpha_, linestyle="--", color="k", label="alpha: CV estimate"
)

plt.legend()
plt.xlabel("alphas")
plt.ylabel("Mean square error")
plt.title("Mean square error on each fold")
plt.axis("tight")

ymin, ymax = 60, 80
plt.ylim(ymin, ymax);
In [ ]:
# saving the lasso reg model
ts = time.time()
file = '../models/lassp_reg_model.sav' + datetime.datetime.fromtimestamp(ts).strftime('_%Y-%m-%d-%H_%M_%S') + ".sav"
# pickle.dump(lasso_best, open(file, 'wb')) # uncommented to reduce dumping models, after each reload. 
file
Out[ ]:
'../models/lassp_reg_model.sav_2023-01-16-00_54_12.sav'

"Model 4": Always predict the mean value¶

In [ ]:
y_prediction_model4 = np.full(len(y_test),y_train.mean())
y_prediction_model4
Out[ ]:
array([4.65334148, 4.65334148, 4.65334148, ..., 4.65334148, 4.65334148,
       4.65334148])
In [ ]:
# Mean squared error model 4
model4MSE = mean_squared_error(y_test, y_prediction_model4)
model4MSE
Out[ ]:
64.91328211074314
In [ ]:
# Root mean squared error model 4
model4RMSE = mean_squared_error(y_test, y_prediction_model4, squared=False)
model4RMSE
Out[ ]:
8.056877938180715
In [ ]:
# Mean absolute error model 4
model4MAE = mean_absolute_error(y_test, y_prediction_model4)
model4MAE
Out[ ]:
5.451782561442774

Conclusions¶

Out of the 3 Models we evaluated only Linear regression and K-nearest neighbour were able to beat our "Dummy model" were we always predicted the mean value of the training set. Both Linear Regression and Lasso Regression performed essentially the same and only slightly better than the Dummy Model. It seems that with the given variables we were not able to create a model which can predict yards gained in a football game reliable. This reflects the unpredictable nature of the game, which makes it enjoyable by millions of fans.

In [ ]:
comparison_models_content = {
    "Model":[
        "Linear Regression",
        "K-nearest neighbour",
        "Lasso Regression",
        "Mean Value",
    ],
    "MSE":[
        model1MSE,
        model2MSE,
        model3MSE,
        model4MSE
    ],
    "RMSE":[
        model1RMSE,
        model2RMSE,
        model3RMSE,
        model4RMSE
    ],
    "MAE":[
        model1MAE,
        model2MAE,
        model3MAE,
        model4MAE
    ]
}

comparison_models = pd.DataFrame(comparison_models_content)
comparison_models
Out[ ]:
Model MSE RMSE MAE
0 Linear Regression 63.521652 7.970047 5.315079
1 K-nearest neighbour 97.914741 9.895188 6.549962
2 Lasso Regression 63.423943 7.963915 5.312489
3 Mean Value 64.913282 8.056878 5.451783